A BRIEF INTRODUCTION OF DIFFERENTIAL EQUATIONS
Aerodynamic Analyses of Curved Ball
金力 5070209313
刘思 5070209419
莫品西 5070209420
沈晋 5070209422
Curved ball is quite common in football games. When a football is kicked out, it tends
to fly along a curved path in stead of a straight one. This is mainly because of the
aerodynamic force acting upon the ball. This article discusses the aerodynamic
analyses of curved ball.
Physical Background
A football makes a curved trace because of its interaction with air. If a football flies in
an empty space, it will make only a simple trajectory without any change in its
advancing direction. On the other hand, if a football does not spin horizontally, i.e.
only makes forward or back spin, it will make no curved trace either. Therefore, it is
the horizontal spin (or spin along the z-axis) that give rise to a curved path.
A spinning football interacts with air over its surface. In a microscopic view, air
consists of huge amounts of gas particles which keep moving in a random manner.
When a football is spinning in air, it interacts with the air particles around it. The
spinning ball causes changes in the direction and magnitude of velocity of the
particles. The motion of a single particle is, however, extremely awkward to be
determined. According to theories in fluid dynamics, a fluid field can be introduced to
describe the properties and behaviors of air.
In the analyses of this problem, a fluid field describing the air around a spinning
ball is built based on some assumption and approximation. Next, a Bernoulli equation,
which describes the dynamic properties of the fluid field, is set up. This equation
gives the aerodynamic force acting upon the football. Then a set of dynamic equations
are set up for the football according to Newton’s second law. Finally the equations
will be solved to give the track of the ball.
Some Assumptions:
When analyzing how a ball is moving in the air, we find it hard to solve the problem
directly, because the motion is the combined effects of air drag force, gravity, and the
force caused by rotation. So some assumptions are needed to simplify the problem.
1. Viscosity of air is neglected, so the rotational speed of the ball will not change
when it is flying.
2. The ball is considered to rotate along a vertical line, so the curved effect is
displayed on a horizontal plane. On the vertical plane, the ball is still following a
parabolic motion.
3. A ball has different velocity on the surface if it is rotating, that make our analysis
difficult to start. If we cut the ball into pieces, and every piece can be viewed as a
cylinder. Then the analysis will be easier. The forces on the cylinders can be
integrated to obtain the net force applied on the ball.
4. Since the relative small velocity along the vertical direction, the vertical drag
force imposed on the ball is neglected. But the horizontal drag force is
indispensable.
5. Some other assumptions will be mentioned in the paper.
Horizontal Motion Analysis:
Lateral force analysis:
A moving cylinder with no rotation in the air can be viewed as static in a moving flow
of air. And the flow is the combination of two flow modes. One is a uniform flow in
the direction opposite that of its moving, and the other mode is a doublet. As is
learned from Fluid Mechanics, some basic flows have their stream functions. They
can be superimposed if several modes are applied at the same time. So the stream
function can be written as
  Ur sin  
K sin 
r
The first term represents the uniform flow mode and the second term represents the
doublet mode. Since  =0 at the surface of the ball, that is r=a, where a is the radius
of the ball. Then the doublet strength K is equal to Ua2.
If the rotation of the ball is considered, we should add a new term ——the free
vortex mode, whose stream function can be written as   ln r . So the final stream
2
  Ur sin  (1 
function is
where  is the circulation(  =  V ds ).
c

a
)
ln r
2
r
2
2
On the surface of circle (r=a), the radial
and tangential velocity can be written as


and
vrs 
0
v s 
r

 2U sin  
r a

2 a
Since we have known the velocity distribution, the pressure distribution on the
surface can be obtained from the Bernoulli equation written from a point far from the
ball where the pressure is p0 and the velocity is U.
1
1
p0  U 2  ps   v s 2
2
2
where Ps is the surface pressure. By substituting v s ,
1
U sin   2
ps  p0  U 2 (1  4sin 2  ) 
 2 2
2
a
8 a
Thus, the force distribution can be obtained by integrating on the surface. The lateral
force Fn is integrated as
Fn   
2
0
ps sin  ad   
2
0
p0 sin  ad  
 0  0  U   0   U   0
2
0
2
2
2 U sin 
2  sin 
1
U 2 (1  4sin 2  ) sin  ad  
d  
d
0
0

2
8 2 a
The nonzero lateral force symbolizes that the ball will change its direction as it is
moving, and the force is always perpendicular to the moving direction. Its simple
physical model is shown below.
Figure 1. Free body diagram of the spinning football.
Air drag force analysis:
The drag force is caused by air, whose direction is always opposite that of its motion.
Assume the drag force of air is proportional to the velocity, then
F   kv  ma  m
dv
dt
dv
k
  dt
v
m
k
 t
v  Ue m
,
where U is the initial velocity of the football
The velocity is decreasing with time t because of the effect of air drag force.
Then the force perpendicular to the velocity direction (lateral force)
Fn  U e
k
 t
m
,
and the force opposite to the velocity(drag force)
Fd  kV  kUe
k
 t
m
Since we have derived all the forces and velocities we need, we can trace the track of
the ball.
d 2x
dt 2
d2y
Fn cos   Fd sin   m 2 ,
dt
dy
tan  
dx
 Fn sin   Fd cos   m
and
k
 t
dx 2 dy 2
m
( )  ( )  V  Ue ,
dt
dt
dy
dt
dy
dx
dx
dt
sin  
 dt k , cos  
 dt k
 t
 t
dx 2 dy 2
dx
dy
Ue m
( ) ( )
( ) 2  ( ) 2 Ue m
dt
dt
dt
dt
Substitute sin  and cos  , we get
dy
dx
d 2x
k
m 2
dt
dt
dt
2
dx
dy
d y
  k  m 2
dt
dt
dt
 
Assume
u
dx
dy
,v 
, then
dt
dt
du
dt
dv
u  kv  m
dt
 v  ku  m
d 2 u 2k

dt 2 m
, 2
d v 2k

dt 2 m
du ( ) 2  k 2
du (0)
kU

u  0, x(0)  0, u (0)  U ,

2
dt
m
dt
m
2
2
dv ( )  k
dv(0) U 
v  0, y (0)  0, v(0)  0,


2
dt
m
dt
m
The density of air   1.3kg / m3 ,   2 a 2  2 (0.452 )(6.2)  8m3 / s ,
k  0.5kg / s , m  0.4kg ,
d 2u
du
du (0)
 2.5  678u  0, x(0)  0, u (0)  U  28m / s,
 35m / s 2
2
dt
dt
dt
2
d v
dv
dv(0)
 2.5  678v  0, y (0)  0, v(0)  0,
 728m / s 2
2
dt
dt
dt
so
u (t )  28e 1.25t cos1.53t
v(t )  475.8e 1.25t sin1.53t
x(t )  7.17e 1.25t (1.53cos1.53t  1.25sin1.53t )  10.98
y (t )  121.89e 1.25t (1.53sin1.53t  1.25cos1.53t )  152.37
The horizontal track is shown below.
Figure 2 Horizontal track.
Vertical motion analysis:
Since we do not consider the vertical rotation and vertical drag force, the ball is under
the force of only gravity. Its track is a parabolic curve. The combined motion track is
shown below.
Figure 3 The three-dimensional trajectory.
Recommendation
The main disadvantage of our analyses is that viscosity has not been accounted for. If
the effect of viscosity of air is significant, the Navier-Stokes equations instead of
Bernoulli equation should be applied to analyze the fluid field. Moreover, drag force
in the vertical direction has been neglected in our analyses. Other factors such as wind
or rain are not considered, either.
The motion track figure by Matlab indicates that the major part of our analysis is
correct, because the shape of the curve is reasonable. We can see that the ball goes
within the track which is familiar with our experience. However, there is still some
defect in the result. The actual distance of a ball kick is far less than what we have
calculated. The reason lies in those rough estimations of some parameters of the
model. Moreover, a flying football usually makes back- or forward-spin, which affect
the trajectory as well. In our analyses, such spin has been neglected. If more accurate
parameters were available, and more accurate aerodynamic equations were
established, a better solution might be obtained.